Quoted: I know you have to be a WA resident to buy a handgun in WA. I know you do not have to be a WA resident to buy a rifle in WA. What about shotguns?

Under WA laws a non-resident can purchase a shotgun from a licensee in WA.

Long gun sales to a non resident must be done through an FFL.

Handgun sales must go through an FFL in the buyers home state.

Depending on what state the purchaser is from there may be a a law in their home state against it.  Some states have laws restricting what states it's residents may purchase firearms from.  For example I believe that Colorado limits it's residents to making purchases only in contiguous states.

Duty orders do make a difference.